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Strong laws for weighted sums of i. i. d. random variables. (English) Zbl 1020.60016
Let $(X_n, n\geq 1)$ be a sequence of independent identically distributed random variables satisfying $EX=0$ and $E[\exp (h |X_1|^{\gamma}) ] < \infty $ for any $h>0$ $(\gamma >0)$. Let $(a_{ni}$, $i\leq i \leq n$, $n\geq 1)$ be an array of constants satisfying ${\lim \sup}_{n \to \infty} \frac{1}{n} \sum_{i=1}^n |a_{ni} |^{\alpha} < \infty $ for some $1 < \alpha \leq 2$. Define $b_n = n^{1/\alpha}(\log n)^{1/\gamma}$ for $0 < \gamma \leq 1$ and $b_n = n^{1/\alpha}(\log n) ^{1/\gamma + \delta} $ for $\gamma >1 $, where $\delta =1 -1/\gamma -(\gamma -1)/ (1+ \alpha \gamma- \alpha)$. Then $\frac{1}{b_n} \sum_{i=1}^n a_{ni} X_i \to 0 \text{ almost surely}.$ Also the case $E[\exp (h |X_1|^{\gamma}) ] < \infty $ for some $ h> 0$ is considered.

60F15Strong limit theorems
Full Text: DOI
[1] Bai, Z. D.; Cheng, P. E.: Marcinkiewicz strong laws for linear statistics. Statist. probab. Lett. 46, 105-112 (2000) · Zbl 0960.60026
[2] Choi, B. D.; Sung, S. H.: Almost sure convergence theorems of weighted sums of random variables. Stochastic anal. Appl. 5, 365-377 (1987) · Zbl 0633.60049
[3] Cuzick, J.: A strong law for weighted sums of i.i.d. Random variables. J. theoret. Probab. 8, 625-641 (1995) · Zbl 0833.60031
[4] Hsu, P. L.; Robbins, H.: Complete convergence and the law of large numbers. Proc. nat. Acad. sci. USA 33, 25-31 (1947) · Zbl 0030.20101
[5] Rosalsky, A.; Sreehari, M.: On the limiting behavior of randomly weighted partial sums. Statist. probab. Lett. 40, 403-410 (1998) · Zbl 0937.60014
[6] Stout, W. F.: Almost sure convergence.. (1974) · Zbl 0321.60022
[7] Wu, W. B.: On the strong convergence of a weighted sum. Statist. probab. Lett. 44, 19-22 (1999) · Zbl 0951.60027